2949775127

iv) Putting m — —1 in (2.16), to get inverse moments of k record values from type II exponentiated log-logistic dłstribution as;

p^!r(i-p)(^i+Łtłi

Recurrence relations for inverse moments of gos from (1.2) can be obtained in the fol-lowing theorem.

2.6. Theorem. For type II exponentiated log-logistic distribution and for 2 < r < n, n > 2 k = 1,2,...,

Ocfl^r

^E[X^j ^(r,n,m,k)\ = E[X^j ^(r — l,n,m,k)]

(2.20)

₊ (j ^)⁽⁷^⁺¹ E[X^j~²P(r, n, m, fc)], 0 > j. ap-yr

Proof. The proof is easy.

Remark 2.3: Setting m = 0, k — 1 in (2.20), we obtain a recurrence relation for inverse moments of order statistics for type II exponentiated log-logistic distribution in the form

f¹ —    = e\x³tzI„\ +    -^1x13+1

V af3(n — r+l)J    ^{J 1    1 J} afi(n — r +1)    ^J

Remark 2.4: Putting m = — 1, in Theorem 2.6, we get a recurrence relation for inverse moments of upper k record values from type II exponentiated log-logistic distribution in the form

gp - p)

afik

)ekx;

^lk) r”) =    >)'-"] +    x$_r)y-^w].

3. Relations for product moments

In this Section, the explicit expressions and recurrence relations for single moments of gos and ratio moments of gos are considered. First we need the following Lemmas to prove the main result.

3.1. Lemma. For type II exponentiated log-logistic distribution as giuen in (1.2) and any non-negative integers a, b, c with m^-1

j. (_a o _c) - _ay+J    (-¹)¹'⁺'^,(i/^)(p)(j//³)w

J.Aa,0,c)-a,    _{la(c+1) + p}__um

(3.1)

[a(a + c + 2)+ p + ?-{(< + j)/j8}] ’

Ji,j(a,b,c) = J J x'y^l{F(x)}'‘f(x){h_m(F(y))-h_m(F(x))\^b[F(y)\^cf(y)dydx. (3.2) Proof: From (3.2), we have

where

x^l [F(x)]^a f(x)G(x)dx, y³[E(y)]^c f(y)dy.

(3.3)

(3.4)